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The discrete acyclic convolution computes the 2n-1 sums sum_{i+j=k; (i,j) in[0,1,2,...,n-1]^2} (a_i b_j) in O(n log n) time. By using suitable offsets andsetting some of the variables to zero, this method provides a tool to calculateall non-zero sums sum_{i+j=k; (i,j) in (P cap Z^2)} (a_i b_j) in a rectangle Pwith perimeter p in O(p log p) time.  This paper extends this geometric interpretation in order to allow arbitraryconvex polygons P with k vertices and perimeter p. Also, this extendedalgorithm only needs O(k + p(log p)^2 log k) time.  Additionally, this paper presents fast algorithms for counting sub-cadencesand cadences with 3 elements using this extended method.

Non-Rectangular Convolutions and (Sub-)Cadences with Three Elements